#9018165

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3833 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

(17,18)(19,21)(20,25)(22,32)(23,26)(24,30)(27,29)(28,31)(33,34,40,38)(35,43,47,41)(36,37,42,39)(44,45,48,46);(1,2)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,19,23,22)(18,21,26,32)(20,28,30,27)(24,29,25,31)(33,35,40,47)(34,41,38,43)(36,48,42,44)(37,45,39,46);(2,6)(3,9)(4,12)(14,16)(17,20,23,30)(18,24,26,25)(19,27,22,28)(21,31,32,29)(33,36)(34,39)(35,44)(37,38)(40,42)(41,45)(43,46)(47,48);(1,2)(3,10)(4,8)(5,6)(7,9)(11,16)(12,13)(14,15)(17,20)(18,25)(19,28)(21,31)(22,27)(23,30)(24,26)(29,32)(33,37,40,39)(34,36,38,42)(35,45,47,46)(41,48,43,44);(1,3)(2,7)(4,11)(5,9)(6,10)(8,14)(12,15)(13,16)(17,21)(18,19)(20,29)(22,26)(23,32)(24,28)(25,27)(30,31)(33,38,40,34)(35,41,47,43)(36,37,42,39)(44,45,48,46);(1,4,5,12)(2,8,6,13)(3,11,9,15)(7,14,10,16)(17,22,23,19)(18,21,26,32)(20,27,30,28)(24,29,25,31)(33,39,40,37)(34,42,38,36)(35,46,47,45)(41,44,43,48)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=gh,d^2=g,e^2=g,f^2=i,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8925361);